Question 98
Let $N$ be a nonnegative integer and let $f, g: \mathbb{Z} \to [0, \infty)$ be functions such that $f(n) = g(n) = 0$ for all $|n| \geq N$ where $\mathbb{Z}$ is the set of all integers. Define $h: \mathbb{Z} \to [0, \infty)$ by
$$
h(n) = \max \{f(k)g(n-k) : k \in \mathbb{Z}\}
$$
for all $n \in \mathbb{Z}$. Prove that
$$
\sum_{n \in \mathbb{Z}} h(n) \geq \left( \sum_{n \in \mathbb{Z}} (f(n))^p \right)^{1/p} \left( \sum_{n \in \mathbb{Z}} (g(n))^q \right)^{1/q}
$$
for all positive real numbers $p$ and $q$ satisfying $1/p + 1/q = 1$.
$$
h(n) = \max \{f(k)g(n-k) : k \in \mathbb{Z}\}
$$
for all $n \in \mathbb{Z}$. Prove that
$$
\sum_{n \in \mathbb{Z}} h(n) \geq \left( \sum_{n \in \mathbb{Z}} (f(n))^p \right)^{1/p} \left( \sum_{n \in \mathbb{Z}} (g(n))^q \right)^{1/q}
$$
for all positive real numbers $p$ and $q$ satisfying $1/p + 1/q = 1$.